Let A be the set of all composites less than 10, and B be the set of positive even integers less than 10. How many different sums of the form a + b are possible if a is in A and b is in B?

Question

Eva Adamson · Accepted Answer

16 different forms of #a+b#. 10 unique sums.

The set #bb(A)# A composite is a number that can be divided evenly by a smaller number other than 1. For instance, 9 is composite #(9/3=3)# but 7 is not (another way of saying this is a composite number is not prime). This all means that the set #A# consists of: #A={4,6,8,9}# The set #bb(B)# #B={2,4,6,8}# We're now asked for the number of different sums in the form of #a+b# where #a in A, b in B#. In one reading of this problem, I'd say there are 16 different forms of #a+b# (with things like #4+6# being different than #6+4#). However, if read as "How many unique sums are there?", perhaps the easiest way to find that is to table it out. I'll label the #a# with #color(red)("red")# and #b# with #color(blue)("blue")#: #(("",color(blue)2,color(blue)4,color(blue)6,color(blue)8),(color(red)4,6,8,10,12),(color(red)6,8,10,12,14),(color(red)8,10,12,14,16),(color(red)9,11,13,15,17))# And so there are 10 unique sums: #6, 8, 10, 11, 12, 13, 14, 15, 16, 17#

Abigail Allen · Answer

undefined First, let's identify the sets A and B:
A = {4, 6, 8, 9} (composites less than 10)
B = {2, 4, 6, 8} (positive even integers less than 10)

To find the possible sums of the form a + b where a is in A and b is in B, we can simply find all the combinations of elements from sets A and B and compute their sums.

The possible sums are:
4 + 2 = 6
4 + 4 = 8
4 + 6 = 10
4 + 8 = 12
6 + 2 = 8
6 + 4 = 10
6 + 6 = 12
6 + 8 = 14
8 + 2 = 10
8 + 4 = 12
8 + 6 = 14
8 + 8 = 16
9 + 2 = 11
9 + 4 = 13
9 + 6 = 15
9 + 8 = 17

There are 15 different possible sums of the form a + b.

Kennedy Adams · Answer

undefined The set A contains composites {4, 6, 8, 9}, and the set B contains positive even integers {2, 4, 6, 8}. The possible sums of the form a + b are: 
4 + 2 = 6,
4 + 4 = 8,
4 + 6 = 10,
4 + 8 = 12,
6 + 2 = 8,
6 + 4 = 10,
6 + 6 = 12,
6 + 8 = 14,
8 + 2 = 10,
8 + 4 = 12,
8 + 6 = 14,
9 + 2 = 11,
9 + 4 = 13,
9 + 6 = 15,
9 + 8 = 17.
So, there are 14 different possible sums.